Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 6 + 12x + 3 x^2 - 2 x^3 \]
\[f'\left( x \right) = 12 + 6x - 6 x^2 \]
\[ = - 6 \left( x^2 - x - 2 \right)\]
\[ = - 6 \left( x - 2 \right)\left( x + 1 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow - 6 \left( x - 2 \right)\left( x + 1 \right) > 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x + 1 \right) < 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 2 \right)\left( x + 1 \right) > 0 \Rightarrow \left( x - 2 \right)\left( x + 1 \right) < 0 \right]\]
\[ \Rightarrow - 1 < x < 2 \]
\[ \Rightarrow x \in \left( - 1, 2 \right)\]
\[\text { So },f(x)\text { is increasing on} \left( - 1, 2 \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow - 6 \left( x - 2 \right)\left( x + 1 \right) < 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x + 1 \right) > 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 2 \right)\left( x + 1 \right) < 0 \Rightarrow \left( x - 2 \right)\left( x + 1 \right) > 0 \right]\]
\[ \Rightarrow x <\text{ - 1 or x} > 2 \]
\[ \Rightarrow x \in \left( - \infty , - 1 \right) \cup \left( 2, \infty \right)\]
\[\text { So },f(x)\text { is decreasing on }\left( - \infty , - 1 \right) \cup \left( 2, \infty \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
The function f(x) = xx decreases on the interval
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The slope of tangent at any point (a, b) is also called as ______.
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
